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Graphical Examination of Data. 1.12.1999 Jaakko Leppänen [email protected] Sources. H. Anderson, T. Black: Multivariate Data Analysis, (5th ed., p.40-46) . Yi-tzuu Chien: Interactive Pattern Recognition, (Chapter 3.4) .

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sources
Sources
  • H. Anderson, T. Black: Multivariate Data Analysis,(5th ed., p.40-46).
  • Yi-tzuu Chien: Interactive Pattern Recognition,(Chapter 3.4).
  • S. Mustonen: Tilastolliset monimuuttujamenetelmät,(Chapter 1, Helsinki 1995).
agenda
Agenda
  • Examining one variable
  • Examining the relationship between two variables
  • 3D visualization
  • Visualizing multidimensional data
examining one variable
Examining one variable
  • Histogram
    • Represents the frequency of occurences within data categories
      • one value (for discrete variable)
      • an interval (for continuous variable)
examining one variable1
Examining one variable
  • Stem and leaf diagram (A&B)
    • Presents the same graphical information as histogram
    • provides also an enumeration of the actual data values
examining the relationship between two variables
Examining the relationship between two variables
  • Scatterplot
    • Relationship of two variables

Linear

Non-linear

No correlation

examining the relationship between two variables1
Examining the relationship between two variables
  • Boxplot (according A&B)
    • Representation of data distribution
    • Shows:
      • Middle 50% distribution
      • Median (skewness)
      • Whiskers
      • Outliers
      • Extreme values
3d visualization
3D visualization
  • Good if there are just 3 variables
  • Mustonen: “Problems will arise when we should show lots of dimensions at the same time. Spinning 3D-images or stereo image pairs give us no help with them.”
visualizing multidimensional data
Visualizing multidimensional data
  • Scatterplot with varying dots
  • Scatterplot matrix
  • Multivariate profiles
  • Star picture
  • Andrews’ Fourier transformations
  • Metroglyphs (Anderson)
  • Chernoff’s faces
scatterplot
Scatterplot
  • Two variables for x- and y-axis
  • Other variables can be represented by
    • dot size, square size
    • height of rectangle
    • width of rectangle
    • color
scatterplot matrix
Scatterplot matrix
  • Also named as Draftsman’s display
  • Histograms on diagonal
  • Scatterplot on lower portion
  • Correlations on upper portion
scatterplot matrix cont
Scatterplot matrix (cont…)

correlations

histograms

scatterplots

scatterplot matrix cont1
Scatterplot matrix (cont…)
  • Shows relations between each variable pair
  • Does not determine common distribution exactly
  • A good mean to learn new material
  • Helps when finding variable transformations
scatterplot matrix as rasterplot
Scatterplot matrix as rasterplot
  • Color level represents the value
    • e.g. values are mapped to gray levels 0-255
multivariate profiles
Multivariate profiles
  • A&B: ”The objective of the multivariate profiles is to portray the data in a manner that enables each identification of differences and similarities.”
  • Line diagram
    • Variables on x-axis
    • Scaled (or mapped) values on y-axis
multivariate profiles cont
Multivariate profiles (cont…)
  • An own diagram for each measurement (or measurement group)
star picture
Star picture
  • Like multivariate profile, but drawn from a point instead of x-axis
  • Vectors have constant angle
andrews fourier transformations
Andrews’ Fourier transformations
  • D.F. Andrews, 1972.
  • Each measurement X = (X1, X2,..., Xp) is represented by the function below, where - < t < .
andrews fourier transformations cont
Andrews’ Fourier transformations (cont…)
  • If severeal measurements are put into the same diagram similar measurements are close to each other.
  • The distance of curves is the Euklidean distance in p-dim space
  • Variables should be ordered by importance
andrews fourier transformations cont2
Andrews’ Fourier transformations (cont…)
  • Can be drawn also using polar coordinates
metroglyphs andersson
Metroglyphs (Andersson)
  • Each data vector (X) is symbolically represented by a metroglyph
  • Consists of a circle and set of h rays to the h variables of X.
  • The lenght of the ray represents the value of variable
metroglyphs cont
Metroglyphs (cont...)
  • Normally rays should be placed at easily visualized and remembered positions
  • Can be slant in the same direction
    • the better way if there is a large number of metrogyphs
metroglyphs cont1
Metroglyphs (cont...)
  • Theoretically no limit to the number of vectors
  • In practice, human eye works most efficiently with no more than 3-7 rays
  • Metroglyphs can be put into scatter diagram => removes 2 vectors
chernoff s faces
Chernoff’s faces
  • H. Chernoff, 1973
  • Based on the idea that people can detect and remember faces very well
  • Variables determine the face features with linear transformation
  • Mustonen: "Funny idea, but not used in practice."
chernoff s faces cont
Chernoff’s faces (cont…)
  • Originally 18 features
    • Radius to corner of face OP
    • Angle of OP to horizontal
    • Vertical size of face OU
    • Eccentricity of upper face
    • Eccentricity of lower face
    • Length of nose
    • Vertical position of mouth
    • Curvature of mouth 1/R
    • Width of mouth
chernoff s faces cont1
Chernoff’s faces (cont…)
  • Face features (cont…)
    • Vertical position of eyes
    • Separation of eyes
    • Slant of eyes
    • Eccentricity of eyes
    • Size of eyes
    • Position of pupils
    • Vertical position of eyebrows
    • Slant of eyebrows
    • Size of eyebrows
conclusion
Conclusion
  • Graphical Examination eases the understanding of variable relationships
  • Mustonen: "Even badly designed image is easier to understand than data matrix.”
  • "A picture is worth of a thousand words”
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